If $y=\sin ^{-1} x$, show that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=0$.

Step 1: First Derivative

We are given the function $y = \sin^{-1} x$. To begin, we need to find the first derivative of $y$ with respect to $x$. Recall that the derivative of $\sin^{-1} x$ is $\frac{1}{\sqrt{1-x^2}}$.

Step 2: Rearrange and Square

To simplify the process of finding the second derivative, we can rearrange the first derivative. Multiply both sides by $\sqrt{1-x^2}$ and then square both sides to eliminate the square root. This gives us $(1-x^2)\left(\frac{d y}{d x}\right)^2 = 1$.

Step 3: Second Derivative using Product Rule

Now, we differentiate the equation $(1-x^2)\left(\frac{d y}{d x}\right)^2 = 1$ with respect to $x$. We use the product rule for the left side and the chain rule for $\left(\frac{d y}{d x}\right)^2$. The derivative of the right side (1) is 0.

Step 4: Simplify the Equation

We can factor out $2\frac{d y}{d x}$ from both terms on the left side of the equation. This simplifies the expression and brings us closer to the desired form.

Step 5: Final Proof

Since $\frac{d y}{d x} = \frac{1}{\sqrt{1-x^2}}$ is not equal to zero for $x \neq \pm 1$, we can divide both sides of the equation by $2\frac{d y}{d x}$. This yields the required identity.

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